It is time to prove some
theorems. There are various proof techniques, and this chapter describes the
most straightforward approach, a tech-nique called direct proof.
As we begin,
it is important to keep in mind the meanings of three key terms: Theorem, proof
and definition.
A theorem is a mathematical
statement that is true, and can be (and has been) verified as true. A proof of
a theorem is a written verification that shows that the theorem is definitely
and unequivocally true. A proof should be understandable and convincing to
anyone who has the requisite background and knowledge. This knowledge includes
an understanding of the meanings of the mathematical words, phrases and symbols
that occur in the theorem and its proof. It is crucial that both the writer of
the proof and the readers of the proof agree on the exact meanings of all the
words, for otherwise there is an intolerable level of ambiguity. A definition
is an exact, unambiguous explanation of the meaning of a mathematical word or
phrase. We will elaborate on the terms theorem and definition in the next two
sections, and then finally we will be ready to begin writing proofs.
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